

% \begin{figure*}[t!]
%   \begin{center}
%   \begin{minipage}[t]{0.31\textwidth}
%      \includegraphics*[width=1\textwidth]{fig/ff_flat.eps}
%   \caption{Impact of Femto capacity reduction factor on the Revenue ($N=200,$
%     $\beta = 0.0048,$ $\theta = 0.5,$ $c_f = 0$)}
%   \label{Flo:par}
%   \end{minipage}
%   \hspace{0.3cm}
%   \begin{minipage}[t]{0.31\textwidth}
%      \includegraphics*[width=1\textwidth]{fig/nbeta.eps}
%      \caption{Normalized revenue of open-femto BSs by that of just
%     closed-femto BSs ($\theta = 0.5$, $\eta = 1$). The percentages on the top of
%     bars indicates the revenue increasing by open femto BSs.}
%   \label{Flo:nbeta}
%   \end{minipage}
%   \hspace{0.3cm}
%   \begin{minipage}[t]{0.31\textwidth}
%      \includegraphics*[width=1\textwidth]{fig/thralpha.eps}
%        \caption{Users' subscription ratios in flat pricing ($N=200,$
%     $\beta = 0.0048,$ $\theta = 0.5,$ $\eta =1$)}
%   \label{Flo:throughputflatalpha}
%   \end{minipage}
% \end{center}
% \end{figure*}


\section{Numerical Results}
\label{sec:numerical}

\subsection{Setup}
We now provide numerical results, where in most cases, we plot the
provider's revenue, user surplus, social welfare, and user subscription ratio for
different values of femto costs, pricing schemes, and users'
price-sensitivities. We tested different values and observed similar
trends to those presented in this section. 
% We do not include social welfare
% plots because they can be calculated
% by just adding the provider's revenue and user surplus.

% tested numerical
% results for different values of parameters, where we observed similar
% trends.

%  except for {\em femto cost} and {\em
%   price-sensitivity}.  This is natural, because the economic metrics and
% user/provider's decisions should significantly depend on the those
% cost-related parameters.


%=================================================================
% \begin{table}[ht]
%   \caption{Default Parameter Values in Numerical Results}
%   \begin{centering}
%     \begin{tabular}{|c|c|}
%       \hline
%       Parameters  & Value\tabularnewline
%       \hline
%       \hline
%       $N$ (\# of users/cell)& 100\tabularnewline
%       \hline
%       $C_{M}$ (macro capacity)& 1\tabularnewline
%       \hline
%       $C_{F}$ (femto capacity)& 2\tabularnewline
%       \hline
%       $\bar{\gamma}$ (max. user type)& 1\tabularnewline
%       \hline
%       $\delta_o$  (prob. of being outside) & 0.6\tabularnewline
%       \hline
%       $\beta$  (femto's relative coverage) & 0.01\tabularnewline
%       \hline
%      $\theta$ (price sensitivity) & 0.5 (default)\tabularnewline
%       \hline
%    \end{tabular}
%     \label{tab:simparameter}
%     \par\end{centering}
% \end{table}

We consider a cellular network with N users/cells, where $N$ is tested
ranging from 100 to 350. Both $C_M$ and $C_F$ are set equal to 1.  Note
that the actual numbers of $C_M$ and $C_F$ are not critical, because
revenue, user surplus, and social welfare just scale with those numbers;
our main interest lies in investigating the relative ratios and changes
of the metrics. The $C_F$ can vary according to the transmission rate of
the backbone network or the transmission power level of the femto
BS. However, the ratio of $C_M$ to $C_F$ seems realistic, because the
power level of femto BSs is set to give the same SINR to users on the
boundary between femtocells and macrocells \cite{CHS08OFC}.  The
probability of users being inside is set to be 0.4 \footnote{According
  to \cite{cisco_data}, the probability of being at home is 40\% and
  being at work is 30\%, roughly. As our focus is on the individual
  user's acceptance of femto BSs, we use the probability of being home
  as that of being inside.}.  The value $\beta$, the coverage of femto
BSs (normalized by that of a macro BS) is tested over $[0.0048,0.03]$,
where we use 0.0048 unless explicitly mentioned. This value is obtained
for macro and femto cells with radiuses of 500 m and 20 m, respectively,
and macrocells exploit a three-sector topology. The value of maximum
user type, $\bar{\gamma}$ is set to 1. For all simulations, unless
explicitly mentioned, the price sensitivity is chosen as 0.5, which is
the median of the interval $[0,1].$ We vary the femto capacity reduction
factor $\eta$ in the interval $[0.2,1],$ where we use $\eta=1$ unless
explicitly mentioned.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}[t]
  \begin{center}
    \subfigure[Revenue]{
      \includegraphics[width=0.6\textwidth]{fig/rcompare_moc.eps}
    }
    \subfigure[User surplus]{
      \includegraphics[width=0.6\textwidth]{fig/scompare_moc.eps}
    }
     \subfigure[Social welfare]{
       \includegraphics[width=0.6\textwidth]{fig/wcompare_moc.eps}
    }
  \end{center}
  \caption{Flat pricing: value-added of the femto services ($N=200,$
    $\beta = 0.0048,$ $\theta = 0.5,$ $\eta = 1$)}
  \label{Flo:throughputopenrev}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}[t]
  \begin{center}
    \subfigure[Revenue]{
      \includegraphics[width=0.6\textwidth]{fig/volume_cr.eps}
    }
    \subfigure[User surplus]{
      \includegraphics[width=0.6\textwidth]{fig/volume_cs.eps}
    }
    \subfigure[Social welfare]{
      \includegraphics[width=0.6\textwidth]{fig/volume_cw.eps}
    }
  \end{center}
  \caption{Partial volume pricing: value-added of the femto services ($N=200,$
    $\beta = 0.0048,$ $\theta = 0.5$, $\eta =1$). }
  \label{Flo:volumecost}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage

\subsection{Value-added of the Open-Femto Service} \label{sec:exp}

Figs.~\ref{Flo:throughputopenrev} and \ref{Flo:volumecost} show the
impact of {\em open-femto} services on the revenue and user surplus,
when $\eta =1.$
 We compare three different cases: {\em
  1)} no femto, {\em 2)} only with closed-femto BSs, and {\em 3)} with
both closed and open femto BSs. We first observe that both revenue and
user surplus increase with the introduction of {\em closed-femto}
services for all pricing schemes, as also reported in
\cite{walrand}. From these results, we can compute social welfare,
which also increases thanks to the {\em closed-femto} service, because
social welfare is the sum of revenue and user surplus.  The
introduction of {\em open-femto} services, %although open and closed
%femto BSs' capacity is the same when $\eta =1$, 
further increases
revenue and user surplus because opening femtocells increases the
total capacity of the system~\footnote{It is called {\em positive
    externality} in economics.}. 

\begin{figure}[t!]
  \begin{center}
    \includegraphics[width=0.7\columnwidth]{fig/ff_flat.eps}
  \end{center}
  \caption{Impact of Femto capacity reduction factor on the Revenue ($N=200,$
    $\beta = 0.0048,$ $\theta = 0.5,$ $c_f = 0$)}
  \label{Flo:par}
\end{figure}

Fig.~\ref{Flo:par} shows the impact of interference between femto BSs
and the macro BS. The $x$-axis and $y$-axis correspond to the
interference factor and the revenue, respectively.  The revenue with the
open femto services is constant while that with the closed femto service
decreases as $\eta$ decreases. The differences between the open femto
and the closed femto increase even more as $\eta$ decreases, which shows
the superiority of the open femto policy.  As more macro users interfere
femto BSs, the value of closed-femto BSs decreases linearly.



\begin{figure}[t!]
  \begin{center}
      \includegraphics*[width=0.7\columnwidth]{fig/nbeta.eps}
  \end{center}
  \caption{Normalized revenue of open-femto BSs by that of just
    closed-femto BSs ($\theta = 0.5$, $\eta = 1$, $c_f =0$). The percentages on the top of
    bars indicates the revenue increasing by open femto BSs.}
  \label{Flo:nbeta}
\end{figure}

The value-added of open femtocells changes depending on the coverage
$\beta$ of open femto BSs and the number of users $N$. In
Fig.~\ref{Flo:nbeta}, the normalized revenue of open-femto BSs by that
of just closed-femto BSs is monotonically increasing, as $\beta$ and
$N$ increase. This is because more offloading can be achieved by
open-femto BSs than closed-femto BSs.

\begin{figure}[t!]
  \begin{center}
      \includegraphics*[width=0.7\columnwidth]{fig/thralpha.eps}
  \end{center}
  \caption{Users' subscription ratios in flat pricing ($N=200,$
    $\beta = 0.0048,$ $\theta = 0.5,$ $\eta =1$)}
  \label{Flo:throughputflatalpha}
\end{figure}

The value-added of open femto services decreases with increasing femto
costs.  This reduction is because more users select the {\em
  macro-only} service with high femto costs, as also shown in
Fig.~\ref{Flo:throughputflatalpha}.  In our environments, for femto
costs higher than 0.4, no value-added is observed.  Note that the cost
0.4 is very high in that 0.13 is the price that maximizes the revenue
without femto BSs, that is, the cost 0.4 is approximately three times
larger than the price of the macro-only service.
Theorem~\ref{thm:femto} supports the observations above, stating that
when the femto-cost is not significantly high, the open-femto service
generates higher revenue than the macro-only service under flat
pricing does.  For a given femto cost $c$, $R_M(c),$ $R_A(c)$ and
$R_F(c),$ denote the maximum revenues with no {\em open-femto}
service, {\em open-femto} with {\em open-to-all}, and {\em open-femto}
with {\em open-to-femto,} respectively.
\begin{theorem}
\label{thm:femto}
Under flat pricing, there exist numbers $\bar{c}_A$ and $\bar{c}_F$ such
that $R_A(c) \geq R_M(c)$ for any $0 \le c \le \bar{c}_A$ and $R_F(c)
\geq R_M(c),$ for any $0\le c \le \bar{c}_F.$ Moreover,
$\bar{c}_A > \bar{c}_F.$
\end{theorem}

The proof and detailed expressions of $\bar{c}_A$ and $\bar{c}_F,$ are
presented in the Appendix.  In order to offer practical insight, we
numerically computed $\bar{c}_A$ and $\bar{c}_F$ for $\theta=0.5$:
$\bar{c}_A=0.38$ and $\bar{c}_F=0.36.$ Thus, in flat pricing, the {\em
open-to-all} policy is more economically robust to the femto cost, and, in
our environments, it is verified that the bounds in
Theorem~\ref{thm:femto} are a good match for those from the numerical
computation.


Fig.~\ref{Flo:roi} shows the Return on Investment (RoI) of femto BSs
over femto costs for both pricing schemes. RoI intuitively refers to
the increase in revenue compared to the invested capital. We assume
that the investment is proportional to the number of deployed femto
BSs. Thus, we define RoI as $(R - R_{\text{macro}})/(\alpha_o +
\alpha_c)N,$ where $R$ and $R_{\text{macro}}$ denote operator revenue
and the revenue only with macro BSs, respectively.  In practice, RoI
is a useful metric for making decisions on investment, which gives an
insight into how long it will take to recover the investment to the
femtocell services \cite{CAG08FNS}. For both pricing schemes, the
RoIs of the three deployment scenarios are similar (see
Fig.~\ref{Flo:roi}), whereas, in the revenue graph, the revenue of
``with closed femto BS only'' is significantly smaller than other
``with `open to femto' policy''. This is because open femto BSs are
promoted by a subsidy so that the number of femto BSs can increase,
although revenue increases more with open femto BSs.





% In the flat pricing, it seems clear that {\em
%   open-to-all} gives much more revenue to the operator with smaller
% $\eta$, because the open femto BSs under {\em open-to-all} can use all
% frequency resource, whereas the closed femto BSs are restricted. The
% revenue decrease in {\em open-to-femto} is similar to that in {\em
%   closed-femto.}  For partial volume pricing scheme, for high $\eta >
% 0.5,$ the revenue of {\em open-to-all} and {\em closed-femto} is the
% same, because under {\em open-to-all}, the ``free-riding'' of {\em
%   mobile only} users on open femto BSs hinders users from subscribing to
% {\em open-femto} service.  However, as the $\eta$ decreases, the value
% of {\em open-to-all} becomes visible. The capacity incentive of open
% femto BSs in low $\eta <0.4$ generates more revenue in spite of
% free-riding of {\em mobile only} users.

\begin{figure}[t!]
  \begin{center}
   \includegraphics[width=0.7\columnwidth]{fig/roi_fv.eps}
 \end{center}
  \caption{Return on Investment (ROI) of femto BSs ($N=200,$
    $\beta = 0.0048,$ $\theta = 0.5,$ $\eta =1$)}
  \label{Flo:roi}
\end{figure}

\subsection{Impact of Femto Cost on User Behavior, Price, and Subsidy}

The operator may want to provide a subsidy to motivate femto users to
open their femto BSs since the utility of {\em closed-femto} service
always exceeds that of {\em open-femto}.  We define subsidy as the
fraction $\frac{p_c -p_o}{p_c}.$ This definition is adopted to reflect
the fact that since open-femto users induce positive externalities,
operators discount subscription fees for the femto service.  Then,
interesting questions include (i) how much subsidy is necessary, and
(ii) how may users will subscribe to each service for the given subsidy.

Fig.~\ref{Flo:throughputflatalpha} shows how the user behavior changes
when femto costs or femto open policies vary under flat pricing.  We
observe that when the femto cost is low ($c_f \leq 0.2$), the majority
of users join the {\em open-femto} or {\em closed-femto} services,
whereas the subscription ratio decreases significantly as femto cost
increases ($c_f > 0.2$). As shown in
Fig~\ref{Flo:throughputflatprice}, the provider whose objective is
revenue maximization selects low prices for the {\em mobile-only}
service for high femto cost, in which case, for the provider it is
hard to attract more users subscribing to femto services.

%=================================================================
\begin{figure}[t!]
  \begin{center}
      \includegraphics*[width=0.7\columnwidth]{fig/thrprice.eps}
  \end{center}
  \caption{Prices in open-to-all policy in flat pricing. The percentages
    on top of the bar graphs represent the subsidy, calculated by
    $\frac{p_c - p_o}{p_c}.$ ($N=200,$
    $\beta = 0.0048,$ $\theta = 0.5,$ $\eta=1$)}
  \label{Flo:throughputflatprice}
\end{figure}

%====================================================================


% The amount of subsidy to provide to open-femto users is an interesting
% question.
% In this paper, we define this subsidy as the fraction
% $\frac{p_c -p_o}{p_c}.$ This definition is made to reflect the fact that
% since open-femto users induce positive externalities, operators discount
% subscription fees for femto services.
Our numerical study suggests that
the subsidy ranges between 10\% and 20\%, as shown in
Fig.~\ref{Flo:throughputflatprice}.  We also observe that the provider
may still start an open-femto business despite $(p_o-p_m) < c_f,$ which
implies that it should pay more money to install and maintain a
femtocell than the increased price from introducing femtocells.  This is
illustrated in Fig.~\ref{Flo:throughputflatprice} for the femto cost $>
0.2$ and the {\em open-to-all} policy. Once again, the reason for this
is the positive externalities of open femto BSs. Under the regime of
non-negligible portion of {\em mobile-only} users, the provider can
increase the price $p_m$ and thus increase the revenue earned from the
{\em mobile-only} users.  Despite a sufficient subsidy, the operator
sustains high revenues, because more {\em open-femto} users lead both
femto users and {\em mobile-only} users to increase their utilities and
thus a high price is acceptable to users.

When interference factor $\eta$ is considered, e.g., $\eta <1,$ {\em
  open-femto} users start to appear without any subsidy, since the
larger capacity of open femto BSs provides enough incentive to open
BSs. For example, in our simulation, when $\eta < 0.7,$ no {\em
  closed-femto} users exist without subsidy. Thus, we can conclude that
restrictive use of resource for closed femto BSs due to interference can
be an enough incentive to open.





\subsection{Open-to-all vs. Open-to-femto Policies}

Fig.~\ref{Flo:throughputopenrev} shows that the plots for the two
polices are close for all values of femto costs, where a small
economic gain is observed in the {\em open-to-all} policy over the
cost range $[0.2, 0.4]$.  When the femto cost is less than 0.2, it is
trivial that there is no difference between {\em open-to-all} and {\em
  open-to-femto} because users do not subscribe to the {\em
  mobile-only} service as shown in
Fig.~\ref{Flo:throughputflatalpha}. On the other hand, over the cost
range $[0.2, 0.4]$, {\em mobile-only} and femto service users can
coexist and thereby, {\em mobile-only} users influence the economic
aspects of the system. Over this cost range, the {\em open-to-all}
policy produces only positive effects on user surplus because open
femto BSs offload more data. However, the gain on revenue is minor
because under {\em open-to-all} the price for femto services should be
discounted due to the loss on their utility by the {\em mobile-only}
users' access, whereas the revenue earned from {\em mobile-only} users
increases.

However, the impacts of these open policies on user behavior is
noteworthy.  When $c_f$ is low, user behavior is almost identical
regardless of the policy.  Differences exist only when the femto cost
is high.  Under the {\em open-to-femto} policy, users that subscribe
to the femto service decide not to share their femto services, whereas
under the {\em open-to-all} policy, users choose the {\em open-femto}
service instead of the {\em closed-femto} service. Users seem to have
higher incentives to share their femto BSs under the {\em open-to-all}
policy than they do under the {\em open-to-femto} policy. Because
sharing helps macro users as well as femto users, the operator can
provide enough subsidy to persuade users to share their femto BSs.

Decrease in subsidy when $c_f = 0.4$ does not have a strong impact on
the overall analysis, because for such a high cost, the users of femto
services are extremely small or even disappear. We model the total femto
operational costs as being linearly proportional to the number of femto
users, as seen in (\ref{eq:revenue}). However, this model may not
reflect certain practical cases. Thus, the results for high femto costs,
for example, $c_f > 0.3$ may not deserve much attention.  % We comment
% that in open-to-all policy the actual system capacity may be larger than
% that in open-to-femto policy in practice, because in open-to-femto {\em
%   mobile-only} users close to femto BSs functions as an interferer to
% femto BSs. Thus, the gap of this subsection is possible to increase.








\subsection{Flat vs. Partial Volume}

We now study the impact of pricing schemes. As shown in
Fig.~\ref{Flo:flatvolume}, we observe that in flat pricing the revenue
is no less than that with partial volume pricing over most values of
elasticity parameter, $\theta.$ In particular, smaller $\theta$ results
in a significant gap in revenue between two pricing schemes.
We interpret this result as follows:

In flat pricing, it is widely known that the users with higher
willingness to pay, that is, $\gamma$, tend to dominate the network
resources~\cite{cisco_data,MV95PC}. This ``negative externality'' (i.e.,
congestion) due to users' heterogeneity in terms of willingness to pay
in flat pricing can be alleviated in various ways which includes
QoS-provisioning mechanism, i.e., imposing the maximum rate on the users
with high demands or guaranteeing the minimum rate to the users with
small demands.  Volume pricing can clearly be another solution that lets
the users with higher demand pay more.  Adding QoS control to flat
pricing often leads to larger revenue than volume
pricing~\cite{Altmann2001519}.  In our model, scheduling across users in
a cell is assumed to be fair, and thus each user is served with a
similar rate differently from the actual demand, which behaves like a
QoS-control mechanism mentioned above. Note that this assumption is not
significantly impractical. For example, in Korea, operators still adopt
flat pricing (i.e., unlimited plan), but, with a QoS control which
constrains users' maximum usage per day.


The revenue difference between two pricing schemes becomes larger with
smaller $\theta.$ Note that $\theta$ is an elasticity parameter, i.e., as
$\theta$ goes to 1, the traffic demand significantly varies as the user
type $\gamma$, whereas, as $\theta$ goes to 0, the traffic demand is
insensitive to the user type $\gamma,$ and thus whether users are served
or not itself affects the provider's revenue, not the traffic
demand. From a simple calculation, the traffic demand over a macro BS
under volume pricing, $x^M(\gamma)$, for a given user type $\gamma,$ is
given by $x^M(\gamma) = (\frac{\gamma \theta}{p})^{1/(1-\theta)}.$ Thus,
for low $\theta$, users tend to transmit a (relatively) small volume of
data with volume pricing, even if the price $p_v^M$ is low.  However,
under flat pricing, the maximum price which guarantees a positive
net-utility is $\gamma x^\theta$ (due to the condition of $\gamma
x^\theta - p \geq 0$) and $\gamma x^\theta$ increases as $\theta$
decreases (due to $x<1$).  Thus, users are willing to subscribe to the
service even with high price, which leads to higher revenue in flat
pricing.


In regard to user surplus, we observe that a sharp increase of surplus
for some specific value of $\theta$: $\theta=0.4$ for $c_f = 0$ and
$\theta=0.1$ for $c_f =0.3$ in Fig.~\ref{Flo:flatvolume}. We resort to
Theorem~\ref{thm:pvolume} to interpret this.  For small values of
$\theta$, the LHS of the condition (\ref{con:p}) is likely to be met,
where the provider chooses $p_{v}^{M} = \infty$ to maximize the
revenue. In that case, no users generate data traffic at macro BSs,
which connects large decrease in surplus. As $\theta$ increases, the
condition (\ref{con:p}) starts to be unsatisfied, then users accordingly
start to use macro BSs with the price of (\ref{eq:lbound}) and
experience the increase in surplus. For very high $\theta$ (very low
price-sensitivity), the provider can attract the users with the
increased $p_v^M,$ again reduces the user surplus.


%====================================================================
\begin{figure*}[t!]
  \begin{center}
\subfigure[Revenue]{
    \includegraphics[width=0.6\textwidth]{fig/volume_r.eps}
}
\subfigure[User surplus]{
    \includegraphics[width=0.6\textwidth]{fig/volume_s.eps}
}
\subfigure[Social welfare]{
    \includegraphics[width=0.6\textwidth]{fig/volume_w.eps}
}
 \end{center}
  \caption{Impact of pricing schemes ($N=200,$
    $\beta = 0.0048,$ $\eta =1$)}
  \label{Flo:flatvolume}
\end{figure*}

%=================================================================
\clearpage

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